It's not quite as bad as the above replies say.
The cloud gaming services, game live streaming programs, and more generic remote desktop tools make use of the h.264 encoder built into modern GPUs. They take what's supposed to be displayed on the screen, compress it with h.264 in real time, and transmit that over the network. The receiving computer simply sees a video stream just like a streamed movie. Decompress and display it, and you have a live view of the sending computer's desktop (with a few tens of ms of latency, and compression artifacts).
How tolerable this is depends on your network connection. Steam Link did a pretty good job of it for games, but required a max of 25 Mbps. It could however work with as little as 3 Mbps (though obviously with more severe compression artifacting). It's limited to 1080p and 60 fps though.
Other desktop sharing services like Splashtop do the same thing, and some are designed to work over the Internet in general. Their limitation is usually your home network connection (particularly upload speed). If your home connection is fast enough, you can get tolerable quality out of this. I'm not sure if any of these are capable of streaming 4k desktops yet.
The situation should get better as GPU hardware becomes faster. Most video streaming services have transitioned to h.265, which in my experience offers similar video quality as h.264 for about 70% the bandwidth. But h.265 encoding is still very CPU/GPU intensive, and AFAIK can't be done yet in real-time.